W-WIMP Annihilation as a Source of the Fermi Bubbles

35 Pages·2013·4.79 MB·English

W-WIMP Annihilation as a Source of the Fermi Bubbles
Luis Alfredo Anchordoqui∗ and Brian James Vlcek†
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1
Department of Physics,
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2 University of Wisconsin-Milwaukee,

g Milwaukee, WI 53201, USA
u
A (Dated: May 2013)
Abstract
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2
The Fermi Gamma-ray Space Telescope discovered two γ-ray emitting bubble-shaped structures

] that extend nearly symmetrically on either side of our Galaxy and appear morphologically con-
h
p nected to the Galactic center. The origin of the emission is still not entirely clear. It was recently
-
p shown that the spectral shape of the emission from the Fermi bubbles is well described by an ap-
e proximately 50 GeV dark matter particle annihilating to b¯b, with a normalization corresponding to
h
[ avelocityaverageannihilationcrosssection(cid:104)σbv(cid:105) ≈ 8×10−27cm3/s. Westudytheminimalhidden

sector recently introduced by Weinberg and examine to what extent its weakly interacting massive
6
v particles (W-WIMPs) are capable of accommodating both the desired effective annihilation rate
5
into quarks and the observed dark matter density.
2
6
4
.
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0
3
1
:
v
i
X
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
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I. INTRODUCTION
Beyond standard model (SM) physics models to be probed at the Large Hadron Collider
(LHC) often include the concept of a hidden sector, consisting of SU(3) × SU(2) × U(1)
singlet fields. Independent of any model, the standard sector and the hidden sector are
coupled by interactions of gauge-invariant operators which illuminate the path for exploring
structures in the hidden sector by observing phenomena in the visible standard sector. A
tantalizing realization of this idea is provided by the Higgs portal, which connects the Higgs
fields in the two sectors by an elementary quartic interaction [1–8]. Such a construct moves
a precision study of the Higgs sector into a central position of new physics searches at the
LHC. Likewise, astrophysical observations open the gates for complementary information to
further test the Higgs portal hypothesis and to improve our understanding of the physics in
the hidden sector.
Around the Galactic center (GC), there exists a bright and spatially extended source
γ-ray emission peaking at energies of a few GeV. The spectrum and morphology of this
signalisconsistentwithoneoriginatingfromdarkmatterannihilations[9–12]. Veryrecently,
evidence of this signal has been found from regions outside of the GC [13] in the directions
of the sky coincident with the Fermi bubbles: two bilateral γ-ray lobes centered at the
core of the Galaxy and extending to around 50◦ above and below the Galactic plane (i.e.,
r = ±10 kpc, where r is the distance from the GC) [14, 15]. At lower Galactic latitudes,
these structures are coincident with a nonthermal microwave “haze” found in WMAP 23−
33 GHz data [16] (confirmed recently by the Planck space mission [17]) and the thermal
x-ray emission seen by ROSAT [18].
Far from the Galactic plane (|b| (cid:38) 30◦), the observed energy-weighted γ-ray spectrum
is nearly invariant with latitude and fairly flat (dΦ /dE ∝ E−2) over the energy range
γ γ γ
observed by Fermi. The correlation found in the multiwavelength observations seems to
indicate that the bubbles (measured in the range of E ∼ 1−100 GeV) are produced by a
γ
population of GeV−TeV electrons (with an approximately power-law spectrum dΦ /dE ∝
e e
E−3) via inverse Compton scattering of ambient low-energy photons, as the same electrons
e
can also simultaneously produce radio synchrotron radiations in the presence of magnetic
fields [14, 19]. The transparency of this elementary and self-consistent framework provides
strong support for a leptonic origin of the high-latitude emission from the Fermi bubbles.
Conversely, at latitudes closer to the disk (|b| (cid:46) 20◦), the spectrum of the emission
correlated with the bubbles possesses a pronounced spectral feature in E2dΦ /dE peaking
γ γ γ
at E ∼ 1−4 GeV, which cannot be produced by any realistic spectrum of electrons [13].
γ
This implies that a second (non-inverse-Compton) emission mechanism must be responsible
for the bulk of the low-energy, low-latitude emission. The spectral shape of this second
component is similar to the one reported from the GC. The intrinsic non-inverse-Compton
emission appears spatially consistent with a luminosity per volume falling approximately
as r−2.4 − r−2.8. As a consequence, the spectral feature visible in the low-latitude bubbles
is most likely the extended counterpart of the GC excess, now detected out to at least
r ∼ 2−3 kpc. Even though millisecond pulsars possess a spectral cutoff at approximately
the required energy, these sources exhibit a spectral shape that is much too soft at sub-GeV
energies to accommodate this signal [20].
The spectrum and angular distribution of the signal is broadly consistent with one pre-
dicted from ∼ 10 GeV dark matter particles annihilating to leptons, or from ∼ 50 GeV
dark matter particles annihilating to quarks, following a distribution similar to, but slightly
2
steeper than, the canonical Navarro–Frenk–White (NFW) profile. In either case, the mor-
phology of the γ-ray signal requires a dark matter distribution that scales approximately as
ρ ∝ r−1.2−r−1.4; that is, the annihilation rate per volume is proportional to the square of
DM
the dark matter density. Such a dark matter distribution is in good agreement with current
observational constraints [21].
For the 10 GeV dark matter candidate, the normalization of the observed signal requires
a velocity average annihilation cross section on the order of
(cid:104)σ v(cid:105) ∼ 2×10−27 cm3/s = 1.7×10−10 GeV−2, (1)
τ
up to overall uncertainties in the normalization of the halo profile [22]. This light
mass scenario has been further invigorated by various observations reported by the
DAMA/LIBRA [23], CoGeNT [24, 25], CRESST [26], and CDMS [27] collaborations, each
of which report signals consistent with a dark matter particle of similar mass. These four
experiments make use of different technologies, target materials, and detection strategies,
but each reports results that are not compatible with known backgrounds but which can be
accommodated by a light dark matter particle with a mass of about 10 GeV and an elastic
scattering cross section with nucleons of 1−2×10−41 cm2 [28–31].
Dark matter particles can elastically scatter with nuclei in the Sun, leading to their
gravitational capture and subsequent annihilation. Electrons and muons produced in such
annihilationsquicklylosetheirenergytothesolarmediumandproducenoobservableeffects.
Annihilations to taus, on the other hand, produce neutrinos which, for a 10 GeV, can be
observed by Super-Kamiokande. For the required branching into τ+τ− of about 10% – as
given by (1) – existing data constrain the dark matter spin-independent elastic scattering
cross section with protons to be less than 4×10−41 cm2 [32, 33].
For the 50 GeV dark matter particle, the normalization of the observed signal requires a
velocity average annihilation cross section on the order of
(cid:104)σ v(cid:105) ∼ 8×10−27 cm3/s = 6.7×10−10 GeV−2. (2)
b
The XENON-100 Collaboration reported a 90% C.L. bound on the elastic scattering cross
section with nuclei of O(10−44 cm2) [34]. A later analysis arrived at alternative conclusions
allowing for a signal of two events with a favored mass of 12 GeV and large error contour
extending to about 50 GeV [35].
It is worthwhile to point out that the bounds from the combined analysis of 10 dwarf
spheroidals [36, 37], galaxy clusters [38], or diffuse γ-ray emission [39, 40] are not sensitive
enough to probe the velocity average annihilation cross sections (1) and (2).
In this paper we study the minimal hidden sector of Weinberg’s Higgs portal model [41],
and we examine to what extent its free parameters can be adjusted to explain the low-
latitude γ-ray emission from the Fermi bubbles. The layout of the paper is as follows. In
Sec. II we outline the basic setting of the model. In Sec. III we review the constraints related
to experimental searches for new physics at the LHC. After that, in Sec. IV we turn our
attentiontotheprospectsfordirectdarkmattersearches. InSec.Vwestudytheconstraints
from cosmological observations. In Sec. VI we present the main results of this work. We
beginbyconstrainingtheparameterspaceintheHiggssectoralongacorrelationoftheFermi
bubbles’ γ-ray signal with the dark matter annihilation cross section into SM fermions. We
then further constrain the parameter space by matching the thermal relic abundance of dark
matter with the value inferred by cosmological observations. Lastly, in Sec. VII we explore
a region of parameter space which cannot accommodate Fermi observations but remains
interesting in itself. In Sec. VIII we summarize our findings.
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II. W-WIMPs
Weinberg’sHiggsportalmodelisbasedonabrokenglobalU(1)symmetryassociatedwith
the dark matter charge W: the number of weakly interacting massive particles (WIMPs)
minus the number of their antiparticles. The hidden sector contains a Dirac field ψ (carrying
WIMP quantum number W = +1) and a complex scalar field (with W = 2, so that its
expectation value leaves an unbroken reflection symmetry ψ → −ψ). All SM fields are
assumed to have W = 0.
The scalar potential consists of the SM component [s], the isomorphic component in the
hidden sector [h], and the quartic interaction coupling between the two sectors with strength
η . The Lagrangian density for the scalar sector reads
χ
L = |∂Φ |2 +|∂Φ |2 +µ2|Φ |2 −λ |Φ |4 +µ2|Φ |2 −λ |Φ |4 −η |Φ |2|Φ |2, (3)
h s h h h h s s s s χ h s
where Φ is the SM scalar doublet and Φ is a complex scalar field. We separate a massless
s h
Goldstone boson field α(x) and a massive radial field r(x) by defining
1
Φ (x) = √ r(x) ei2α(x), (4)
h
2
where r(x) and α(x) are real, with the phase of Φ (x) adjusted to make the vacuum ex-
h
pectation value (VEV) of α(x) zero. The SU(2)×U(1) symmetry of the SM is (of course)
broken by a nonvanishing VEV of the neutral component φ of the scalar doublet,
(cid:18) (cid:19)
1 G±
Φ = √ , (5)
s 2 vφ +φ(cid:48) +iG0
where v (cid:39) 246 GeV. The G fields are the familiar Goldstone bosons, which are eaten by
φ
the vector bosons (i.e. the G± become the longitudinal components of the charged W boson
and G0 becomes the longitudinal component of the Z boson). In terms of real fields the
Lagrangian density (3) takes the form
1 1 µ2 λ η
L = ∂r2 + ∂φ2 +2r2∂α2 + hr2 − hr4 +µ2|φ|2 −λ |φ|4 − χr2|φ|2. (6)
2 2 2 4 s s 2
The U(1) symmetry of W conservation is also broken and r gets a VEV
r(x) = v +r(cid:48)(x), (7)
r
with v real and non-negative.
r
We demand the scalar potential obtains its minimum value at
µ2 λ µ2 λ η
V = − hv2 + hv4 − sv2 + sv4 + χv2v2. (8)
2 r 4 r 2 φ 4 φ 4 r φ
Physically, the most interesting solutions to the minimization of (8),
η
∂ V = −µ2v +λ v3 + χv v2 = 0 (9)
vr h r h r 2 r φ
and
η
∂ V = −µ2v +λ v3 + χv2v = 0, (10)
vφ s φ s φ 2 r φ
4
are obtained for v and v both nonvanishing
r φ
1 (cid:18) η v2(cid:19)
v2 = µ2 − χ r (11)
φ λ s 2
s
and
1 (cid:18) η v2(cid:19)
v2 = µ2 − χ φ , (12)
r λ h 2
h
respectively. To compute the scalar masses, we must expand the potential around the
minima
1
L = (∂r(cid:48))2
2
+ 2v2∂α2 +4v r(cid:48)∂α2 +2r(cid:48)2∂α2
r r
− λ v2r(cid:48)2 −λ v2φ(cid:48)2 −η v v r(cid:48)φ(cid:48) +··· , (13)
h r s φ χ r φ
where the dots indicate 3-point and 4-point interactions, as well as the SM interactions.
There is a mixing term present for r(cid:48) and φ(cid:48). We find the fields of definite mass by diago-
nalizing the mass matrix for r(cid:48) and φ(cid:48). We denote by H and h the scalar fields of definite
masses, m = 125 GeV and m , respectively. After a bit of algebra, the explicit expressions
H h
for the scalar mass eigenvalues and eigenvectors are given by
(cid:113)
m2 = λ v2 +λ v2 − (λ v2 −λ v2)2 +(η v v )2 (14)
h h r s φ s φ h r χ r φ
and
(cid:113)
m2 = λ v2 +λ v2 + (λ v2 −λ v2)2 +(η v v )2, (15)
H h r s φ s φ h r χ r φ
with
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
h cosχ −sinχ r(cid:48)
= , (16)
H sinχ cosχ φ(cid:48)
where χ ∈ [−π/2,π/2] also fulfills
η v v 2η v v
χ φ r χ φ r
sin2χ = = , (17)
(cid:113) m2 −m2
(λsvφ2 −λhvr2)2 +(ηχvrvφ)2 H h
and
λ v2 −λ v2
s φ h r
cos2χ = , (18)
(cid:113)
(λ v2 −λ v2)2 +(η v v )2
s φ h r χ r φ
yielding
η v v
χ r φ
tan2χ = . (19)
λ v2 −λ v2
s φ h r
The Goldstone boson in (13) has to be be renormalized so that it resumes the standard
canonical form. This is achieved through scaling α → α(cid:48) = 2v α, videlicet,
r
1 1 1
2v2∂α2 +4v r(cid:48)∂α2 +2r(cid:48)2∂α2 → ∂α(cid:48)2 + r(cid:48)∂α(cid:48)2 + r(cid:48)2∂α(cid:48)2. (20)
r r 2 v 2v2
r r
5
Adding in the dark matter sector requires at least one Dirac field
f f∗
L = iψ¯γ ·∂ψ −m ψ¯ψ − √ ψ¯cψΦ† − √ ψ¯ψcΦ . (21)
ψ ψ 2 h 2 h
We assign ψ a charge W = 1, so that the Lagrangian is invariant under the global transfor-
mation eiWα. Treating the transformation as local allows us to express ψ as
ψ(x) = ψ(cid:48)(x)eiα(x). (22)
We can now rewrite (21) in terms of ψ(cid:48), α, and r
f f
L = iψ¯(cid:48)γ ·∂ψ(cid:48) −(ψ¯(cid:48)γψ(cid:48))·∂α−m ψ¯(cid:48)ψ(cid:48) − ψ¯(cid:48)cψ(cid:48)r− ψ¯(cid:48)ψ(cid:48)cr, (23)
ψ ψ
2 2
where we have taken f to be real. Once r achieves a VEV we can expand the dark matter
sector to get
i (cid:16) (cid:17)
L = ψ¯(cid:48)γ ·∂ψ(cid:48) +ψ¯(cid:48)cγ ·∂ψc(cid:48) ,
ψ
2
m fv fv
− ψ (cid:0)ψ¯(cid:48)ψ(cid:48) +ψ¯(cid:48)cψ(cid:48)c(cid:1)− rψ¯(cid:48)cψ(cid:48) − rψ¯(cid:48)ψ(cid:48)c,
2 2 2
1
− (ψ¯(cid:48)γψ(cid:48) −ψ¯(cid:48)cγψ(cid:48)c)·∂α,
2
f
− r(cid:48)(cid:0)ψ¯(cid:48)cψ(cid:48) +ψ¯(cid:48)ψ(cid:48)c(cid:1). (24)
2
Note that we have made the Lagrangian explicitly symmetric via relations like
ψc = Cψ¯T (25)
ψ¯cψc = (−ψTC−1Cψ¯T) = ψ¯ψ (26)
ψ¯ γ ·∂ψ = −ψTC−1γC ·∂ψ¯T
c c
= ψTγT ·∂ψ¯T = −(∂ψ¯·γψ) → ψ¯γ ·∂ψ. (27)
In (26) we used the Grassman nature of the spinor fields; in the second line of (27) we used
integration by parts to transfer the derivative onto the ψ field. Similar results can be found
for the other expressions.
Diagonalization of the ψ(cid:48) mass matrix generates the mass eigenvalues,
m = m ±fv , (28)
± ψ r
for the two mass eigenstates
i 1
ψ = √ (ψ(cid:48)c −ψ(cid:48)) and ψ = √ (ψ(cid:48)c +ψ(cid:48)) . (29)
− +
2 2
In this basis, the act of charge conjugation on ψ results in
±
ψc = ψ . (30)
± ±
6
This tells us that the fields ψ are Majorana fermions. The Lagrangian is found to be
±
i i 1 1
L = ψ¯ γ ·∂ψ + ψ¯ γ ·∂ψ − m ψ¯ ψ − m ψ¯ ψ ,
ψ + + − − + + + − − −
2 2 2 2
i
− (ψ¯ γψ −ψ¯ γψ )·∂α(cid:48),
+ − − +
4v
r
f
− r(cid:48)(ψ¯ ψ −ψ¯ ψ ). (31)
+ + − −
2
Wemustnowputr(cid:48) intoitsmassivefieldrepresentation, forwhichtheinteractionsofinterest
are
f sinχ f cosχ
¯ ¯ ¯ ¯
− H(ψ ψ −ψ ψ )− h(ψ ψ −ψ ψ ). (32)
+ + − − + + − −
2 2
This leads to 3-point interactions between the W-WIMPs and the Higgs boson of the SM.
In summary, instead of one Dirac W-WIMP, there are two Majorana W-WIMPs of dif-
ferent masses. However, the heavier W-WIMP will decay into the lighter one by emitting
a Goldstone boson, while the lighter one is kept stable by an unbroken reflection symme-
try. Therefore in this model we can expect that the Universe today will contain only one
type of Majorana W-WIMP, the lighter one w, with mass m equal to the smaller of m .
w ±
Throughout, ∆m = |m −m | = 2|fv | denotes the mass splitting of the W-WIMP states.
+ − r
(The most common variables used in this article are summarized in Table I.)
A cautionary note is worth taking on board at this juncture. It has long been known
that the spontaneous breaking of a global U(1) symmetry has several disconnected and
degenerate vacua (the phase of the vacuum expectation value (cid:104)0|Φ |0(cid:105) can be different in
h
different regions of space, and actually we expect it to be different in casually disconnected
regions), leading to catastrophic domain-wall structures in the early Universe [42, 43]. In
the spirit of [42], it may be possible to introduce a small explicit breaking of the symmetry,
such that the domain walls disappear before dominating the matter density of the Universe,
while leaving (pseudo-)Goldstone bosons and the same dark matter phenomenology.1
The absence of new physics signals at the LHC place constraints on the model. We
discuss this next.
III. CONSTRAINTS FROM COLLIDER EXPERIMENTS
The recent discovery [49, 50] of a new particle with properties consistent with the SM
Higgs boson is without any doubt the most compelling news from the LHC. With the
measurements in various channels, a comprehensive study of the properties of the Higgs-like
state becomes possible and has the potential for revealing whether or not the Higgs sector is
as simple as envisioned in the SM. Since invisible decays reduce the branching fraction to the
(visible) SM final states, it is to be expected that B(H → invisible) is strongly constrained.
Indeed B(H → invisible) is known to be less than about 19% at 95%C.L. [51–54]. Thus, the
mixing of the SM with the hidden sector must be weak. Note that for η (cid:28) 1 the relations
χ
between masses and angles then become
2η v v
m2 ≈ 2λ v2, m2 ≈ 2λ v2, tan2χ ≈ χ r φ , (33)
h h r H s φ m2 −m2
H h
1 Other approaches, if exceedingly fine-tuned, may offer alternative solutions [44–48].
7
TABLE I: Definition of most common variables.
Φ SM scalar doublet
s
Φ Complex scalar field
h
φ Neutral component of the scalar doublet
r Massive radial field
α Goldstone boson
v Vacuum expectation value of φ
φ
v Vacuum expectation value of r
r
H SM Higgs boson
h Hidden scalar
η Quartic interaction coupling between SM and hidden sectors
χ
χ H-h mixing angle
w Lightest W-WIMP
∆m W-WIMP mass splitting
f Coupling between hidden Majorana fermions and complex scalar field
where we have assumed λ v2 > λ v2. For a Higgs width of about 4 MeV, the partial width
s φ h r
for decay into unobserved particles is found to be
Γ < 0.8 MeV. (34)
H→invisible
The phenomenology of a Higgs portal to the hidden sector depends on whether the SM
Higgs particle is lighter or heavier than the new companion. In this study we take m > m .
H h
The decay rate into invisible stuff, Γ , has two distinct contributions: ΓSM
H→invisible H→invisible
and Γ . The former is dominated by H → 2Z → 4ν, with an invisible Z branching
H→hidden
ratio of 4%. The 4ν rate can be predicted from observed decays H → 2Z → 4l. For the
sake of simplicity, hereafter we will omit the contribution of ΓSM . Unless expressly
H→invisible
stated otherwise herein, we assume m +∆m > m /2 and thus H decays (invisibly) into
w H
the hidden sector via three channels: H → 2α(cid:48), H → 2h, and H → 2w. From the event
rates for visible Higgs production and decay channels, we could derive upper bounds on
non-SM admixtures in the wave function of the Higgs boson and on the new three invisible
decay channels. To this end we now compute the decay rates for these three processes.
A. Γ
H→2α(cid:48)
Substituting in (20) r(cid:48) by the field of definite mass, r(cid:48) = hcosχ+Hsinχ, we can write
the Higgs–Goldstone boson interaction term as
1 sinχ cosχ
r(cid:48)∂α(cid:48)2 → H(∂α(cid:48))2 + h(∂α(cid:48))2. (35)
v v v
r r r
Using (35) we write the Feynman rule for interactions of the type H, α(cid:48), α(cid:48) as
2sinχ
−i k ·k(cid:48), (36)
v
r
8
where k(k(cid:48)) is the 4-momentum of the incoming (outgoing) α(cid:48) particle, and the factor of 2
is a symmetry factor, as one can exchange incoming-outgoing α(cid:48) twice. From this 3-point
interaction we can calculate the decay width of the SM Higgs H into 2 Goldstone bosons α(cid:48).
In the rest frame of the Higgs, the differential decay probability per unit time is given by
1 (cid:18)2sinχ (cid:19)2
dΓ = k ·k dQ, (37)
H→2α(cid:48) 1 2
2m v
H r
where
1 d3k d3k
dQ = 1 2 (2π)4δ(m −k −k )δ(3)(k +k )
2! (2π)32k (2π)32k H 1 2 1 2
1 2
(cid:12)
= 1 dΩk1 (cid:12)(cid:12) (38)
16(2π)2(cid:12)
k1=mH/2
isthephasespaceforatwo-bodyfinalstate(thefactorof1/2!isincludedbecauseofidentical
particles in the final state). After some algebra (37) can be rewritten as
dΩ (cid:20)2 sinχ (cid:16)m (cid:17)2(cid:21)2
dΓ = k1 2 H . (39)
H→2α(cid:48) 128 π2m v 2
H r
The partial decay width can now be expressed as
1 (cid:18)sinχ(cid:19)2
Γ = m3 . (40)
H→2α(cid:48) 32π v H
r
For m (cid:29) m and m2 (cid:29) 2η v v , we can use the small angle approximation
H h H χ r φ
sinχ ≈ χ = η v v /(m2 −m2). (41)
χ r φ H h
In this very good approximation the decay width becomes
1 (cid:18) η v (cid:19)2
Γ = χ φ m3 . (42)
H→2α(cid:48) 32π m2 −m2 H
H h
B. Γ
H→2h
We begin by expanding the scalar potential V around the VEVs of r and φ after which
we diagonalize the mass matrix. Together this requires that we expand around the fields
r(x) = v +hcosχ+Hsinχ ,
r
φ(x) = v +Hcosχ−hsinχ , (43)
φ
9
which puts V in the form
1 1
V = m2 H2 + m2h2
2 H 2 h
1
− (η +3(λ +λ )+3(η −λ −λ )cos4χ)H2h2
χ h s χ h s
16
1
− v cosχ[6λ −η +3(η −2λ )cos2χ]Hh2
φ s χ χ s
4
1
− v sinχ[6λ −η −3(η −2λ )cos2χ]Hh2
r h χ χ h
4
1
(cid:0) (cid:1)
− λ cos4χ+η cos2χsin2χ+λ sin4χ H4
s χ h
4
1
(cid:0) (cid:1)
− λ cos4χ+η cos2χsin2χ+λ sin4χ h4
h χ s
4
1
(cid:0) (cid:1)
+ v sinχ(η cos2χ+2λ sin2χ)−v (2λ cos3χ+η cosχsin2χ) h3
φ χ s r h χ
2
1
(cid:0) (cid:1)
− v sinχ(η cos2χ+2λ sin2χ)+v (2λ cos3χ+η cosχsin2χ) H3
r χ h φ s χ
2
1
− (λ −λ +(λ +λ −η )cos2χ)sin2χ Hh3
h s s h χ
4
1
+ (λ −λ +(λ +λ −η )cos2χ)sin2χ H3h
s h s h χ
4
1
+ v sinχ[2(3λ −η )cos2χ+η sin2χ]H2h
φ s χ χ
2
1
+ v (η sinχsin2χ−6λ cosχsin2χ−η cos3χ)H2h. (44)
r χ h χ
2
Since χ < 1 we first expand the potential around χ = 0, and then using (17) we further
expand around η = 0 retaining only the terms first order in η ; this results in
χ χ
1 1
V ≈ m2 H2 + m2h2
2 H 2 h
η λ λ η λ v v η λ v v
− χH2h2 − hh4 − sH4 − χ h r φ Hh3 + χ s r φ H3h
4 4 4 m2 −m2 m2 −m2
H h H h
η v (cid:18) 6λ v2 (cid:19) η v (cid:18) 6λ v2 (cid:19)
− λ v h3 −λ v H3 − χ φ h r +1 Hh2 + χ r s φ −1 H2h .
h r s φ 2 m2 −m2 2 m2 −m2
H h H h
Using (33) we can manipulate this expression to write the scalar potential as
1 1
V ≈ m2 H2 + m2h2
2 H 2 h
η λ λ η λ v v η λ v v
− χH2h2 − hh4 − sH4 − χ h r φ Hh3 + χ s r φ H3h
4 4 4 m2 −m2 m2 −m2
H h H h
m2 m2 η v (cid:18)m2 +2m2(cid:19) η v (cid:18)2m2 +m2(cid:19)
− hh3 − hH3 − χ φ H h Hh2 + χ r H h H2h . (45)
2v 2v 2 m2 −m2 2 m2 −m2
r φ H h H h
Under the approximations taken previously, m (cid:29) m and m2 (cid:29) 2η v v , the relevant
H h H χ r φ
Hhh interaction term results in
η v (cid:18)m2 +2m2(cid:19)
− χ φ H h Hh2. (46)
2 m2 −m2
H h
10